Monitoring and control of machines and automatic lines. Typical technical solutions for automation of technological processes Closed-loop automatic control system

For normal stable operation of nuclear power plant power units, it is necessary to maintain a number of thermal parameters within specified limits. These functions are implemented by automatic control systems for thermal parameters, the reliable, efficient and stable operation of which largely determines the operation of the power unit as a whole.

In total, at one nuclear power plant unit there are about 150 local automatic control systems (regulators), of which approximately 30-35 can be classified as the most important, in the event of failure of which the power unit is usually switched off by protections (level regulators in the steam generator, deaerator, BRU- MV, pressure in the primary circuit, etc.), or the load of the power unit decreases (level regulators in the HPH).

Maintaining parameters manually for a long time is difficult, time-consuming and requires certain skills from operating personnel. Operation and operational maintenance of regulators at a power unit requires personnel to know the basics of the theory of automatic control, principles of operation, design and hardware on which the regulators are implemented.

Automatic control systems are used in cases where it is necessary to change or maintain constant for a long time any physical quantities called controlled variables (voltage, pressure, level, temperature, rotation speed, etc.) characterizing the operation of the machine, technological process or dynamics of a moving object.

Devices that implement these functions are called automatic regulators.

The object of regulation is a machine or installation, the specified operating mode of which must be maintained by the regulator with the help of regulatory bodies. The combination of the regulator and the regulated object is called an automatic control system.

The automatic control system (CAP) based on the “Cascade-2” equipment is made on the basis of microelectronics in instrument design.

Primary transducers of the Sapphire-22 type with strain-sensitive elements, resistance thermometers and thermocouples were used as the main sources of information.

Let's consider the functional diagram of switching on block D07 with the regulator balanced at the current parameter value (Figure 2.4).

The self-balance of the autoregulator to the current value is based on a change in the reference signal. When the switch is positioned “P” (manual mode), pressing the buttons “B” (more) or “M” (less) sets the controller’s setting.

Figure 2.4 – Block diagram of autoregulator self-balance for the current parameter value

When the switch is in position “A” (automatic mode), the output commands of the P27 control unit (minus 24V) are sent to the “ ” or “ ” inputs, causing changes in the output signal of the D07 block. When the regulator is switched on, the influence of the control pulses of block P27 on the integrator stops (the normally closed contacts of the BVR relay open) and the regulator setting remains equal to the value of the process parameter at the time of switching on.

CPS of the VVER-1000 reactor

Tasks that the nuclear reactor control and protection system must solve:

1. Ensuring that the power or other parameter of the reactor changes in the required range at the required speed and maintains the power or other parameter at a certain specified level. Therefore, to ensure this function, special control rods are needed. They are called automatic control bodies (AR).

2. Compensation for changes in nuclear reactivity. Special PPS bodies that perform this task are called compensation bodies.

3. Ensuring the safe operation of nuclear reactors, which can be carried out by nuclear reactors by stopping the fission chain reaction in emergency situations

CPS is intended:

To automatically regulate the power of nuclear reactors in accordance with the power supplied by the TG to the network, or to stabilize the power at a given level;

To launch a nuclear reactor and bring it to power in manual mode;

To compensate for changes in reactivity in manual and automatic mode;

Emergency protection of nuclear reactors;

To signal the reasons for the activation of the AZ;

For automatic shunting of some AZ signals;

To signal faults occurring in the control system;

To signal the position of the nuclear reactor on the main control room and control room, as well as call information about the position of each operational control in the IVS EB SVRK.

The reactor is controlled by influencing the progress of the central rocket engine of fuel nuclei in the core.

The nuclear control system being developed provides a method for introducing solid absorbers in the form of rods. Along with mechanical controls, the introduction of a boric acid solution into the primary circuit coolant is used. Operational power control is carried out by mechanical movement of the executive bodies containing a solid absorber.

Requirements for CPS:

1. To electrical parameters and modes:

The control system is designed for power supply from at least two independent power sources; when one source disappears, the operation of the control system is maintained;

During a long-term shutdown of power supply parameters, false activation of emergency protection (EP) does not occur and the control elements do not move spontaneously;

The control system must ensure the exchange of information with different systems.

2. Towards reliability:

CPS service life is at least 10 years;

MTBF for control functions 10 5 hours;

The unavailability factor for the core functions, requiring nuclear reactor shutdown, is no more than 10 -5;

Average recovery time 1 hour.

3. To the equipment:

The CPS equipment provides the possibility of functional testing, as well as CPS parameters, using control means during preparation for launch, while the nuclear reactor is running without stopping it, without disrupting the functions of the system and the operability of the reactor installation (RP);

Communication lines are designed so that a fire in one line does not lead to the inability to perform functions.

4. To the actuators:

Elimination of spontaneous movement in the direction of increasing reactivity (in the event of a malfunction, loss of power, and so on);

Operating travel speed 20 ± 2 mm per second;

The time for introducing the working bodies into the core is 1.5 - 4 seconds;

The time from issuing the AZ signal to the start of movement is 0.5 seconds;

The working stroke of the control body is 3500 mm.

CPS composition

PTK SGIU-M

PTK AZ-PZ

PTK ARM-ROM-UPZ

Power supply of equipment.

The general task of process control is to minimize (maximize) a certain criterion (cost, energy costs, etc.) while fulfilling the restrictions on technological parameters imposed by regulations.

Since solving this problem for the entire process as a whole is difficult (there are many influencing factors), the entire process should be divided into separate sections, and usually the section corresponds to a completed technological operation that has its own subtask (feed preparation, milk processing, etc.).

For a separate TP, the optimality criterion is easier to establish. This may be a requirement for stabilization of a parameter or a simple calculated criterion. Based on the accepted optimality criterion for a separate technological process, the automation problem is easily formulated. In addition to the optimality criterion, solving this problem requires an analysis of the automation object from the point of view of identifying all significant input and output variables, as well as an analysis of the static and dynamic characteristics of the transmission channels of disturbing and control influences.

Rice. 2.3. Flow control schemes: A- liquid and gaseous media; b- bulk materials; V- environment ratios

Technological processes of the same type (for example, heating processes) may differ in the design of the equipment, the physical and chemical properties of the raw material flows involved, etc. However, they all proceed according to the same laws and are subject to general patterns. The nature of these patterns is primarily determined by which parameter is involved in control. For one class of processes occurring in a typical technological system, a standard automation solution can be developed that is acceptable for a wide range of systems. The presence of a standard solution greatly simplifies the task of building an automated control system.

Typical process parameters that are subject to monitoring and regulation include flow, level, pressure, temperature and a number of quality indicators.

Flow regulation. Flow control systems are characterized by low inertia and frequent pulsation of the parameter.

Typically, flow control is throttling the flow of a substance using a valve or gate; change in pressure in the pipeline due to changes in the rotation speed of the pump drive or the degree of bypass (diversion of part of the flow through additional channels).

The principles of implementation of flow regulators for liquid and gaseous media are shown in Figure 2.3, A, bulk materials - in Figure 2.3, b.

In the practice of process automation, there are cases when it is necessary to stabilize the ratio of the flow rates of two or more media.

In the circuit shown in Figure 2.3, V, flow G 1 - leading and flow - slave, where at- flow ratio coefficient, which is set during the static adjustment of the regulator.

When the leading thread changes G 1 regulator FF proportionally changes the slave flow G 2.

The choice of control law depends on the required quality of parameter stabilization.

Level regulation. Level control systems have the same features as flow control systems. In the general case, the behavior of the level is described by the differential equation

(2.1)

,

where S is the horizontal cross-sectional area of ​​the container; L- level; C in, G out - flow of medium at the inlet and outlet; With arr.- the amount of medium that increases or decreases in the container (can be equal to 0) per unit of time t.

Constancy of the level indicates the equality of the quantities of supplied and consumed liquid. This condition can be ensured by influencing the supply (Fig. 2.4, A) or flow (Fig. 2.4, b) liquids. In the controller version shown in Figure 2.4, V, the results of measurements of liquid supply and flow are used to stabilize the parameter. The liquid level pulse is corrective; it eliminates the accumulation of errors due to inevitable errors that arise when the supply and flow rate change. The choice of control law also depends on the required quality of parameter stabilization. In this case, it is possible to use not only proportional, but also positional controllers.

Pressure regulation. Constancy of pressure, as well as constancy of level, indicates the material balance of the object.

(2.2)

In the general case, the change in pressure is described by an equation similar to formula (2.1),

Where V- apparatus volume; p - pressure.

Rice. 2.4. Schemes of level control systems:

A-with impact on the feed; b And V- with impact on medium flow

The similarity of equations (2.1) and (2.2) indicates that the methods of pressure regulation are similar to the methods of level regulation.

Temperature regulation. Temperature is an indicator of the thermodynamic state of the system. The dynamic characteristics of the temperature control system depend on the physical and chemical parameters of the process and the design of the apparatus. A feature of such a system is the significant inertia of the object and often the measuring transducer.

The principles for implementing temperature controllers are similar to the principles for implementing level controllers (Fig. 2.4), taking into account the control of energy consumption in the facility.

The choice of the control law depends on the inertia of the object: the greater it is, the more complex the control law. The time constant of the measuring transducer can be reduced by increasing the speed of the coolant, reducing the thickness of the walls of the protective cover (sleeve), etc.

Rice. 2.5. Scheme of the product quality control system:

1 - an object; 2 - quality analyzer; 3 - extrapolation filter; 4 - computing device; 5 - regulator

Regulation of parameters of the composition and quality of the product. When regulating the composition or quality of a product, a situation is possible when a parameter (for example, grain moisture) is measured discretely. In this situation, information loss and a decrease in the accuracy of the dynamic control process are inevitable. Recommended circuit of a regulator that stabilizes some intermediate parameter У(t), the value of which depends on the main regulated parameter - the product quality indicator U( t), shown in Figure 2.5. computing device 4, using a mathematical model of the relationship between parameters У(t) And У(t 1) continuously evaluates the quality indicator. Extrapolation filter 3 gives an estimated product quality parameter У(t 1)in the intervals between two dimensions.

Test questions and assignments

1. Describe the technological process of agricultural production. 2. Name the types of impacts on the control object. 3. Outline the structure and principles of TP management. 4. What are the features of automation of agricultural production? 5. Name typical technical solutions for process automation.

Send your good work in the knowledge base is simple. Use the form below

Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

Posted on http://www.allbest.ru/

Ministry of Education and Science of the Russian Federation

Branch of the federal state budgetary educational institution of higher professional education

"Samara State Technical University" in Syzran

Department of Electromechanics and Industrial Automation

Course project

in the discipline "Design of automated systems"

Regulation of technological parameters at the EOLU AVT-6 installation

Completed:

Student gr. EABZ-401 Golotin K.O.

Checked:

Art. teacher Shumilov E.A.

Syzran 2014

Introduction

1. Description of the installation

3. Regulator calculations

Conclusion

Introduction

Oil has been known to man since ancient times. For centuries, oil has been used as a medicine, fuel, and lighting material. As technology developed in Russia, the oil refining industry also developed, which ensured the production of various petroleum products from oil. The oil industry faces a huge challenge: providing raw materials and intermediate products to the chemical and petrochemical industries. The raw materials for the development of these industries are natural and associated gas, liquefied gas and individual hydrocarbon fractions. In addition, oil refineries began to produce aromatic hydrocarbons, carbon black raw materials, synthetic fatty acids and alcohols, as well as many other products. The modern oil refining industry is constantly under the sign of scientific and technical developments. The main technological processes at oil refineries are: desalting and dehydration of oil at the primary stage, catalytic cracking, catalytic reforming, isomerization, hydrogenation purification of petroleum distillates, etc. - at the secondary and subsequent stages.

The widespread use of secondary oil refining processes increases the requirements for precise oil separation and deeper selections. Modern technological processes of oil refining are characterized by high productivity, high flow rates and certain parameter values, the deviation of which is allowed only within the smallest limits.

The modern world market places high demands on the quality of oil and petroleum products, so it is necessary to continuously improve the quality of products. And this requires the use of modern high-precision control systems.

Oil distillation processes are carried out in so-called atmospheric tubular (AT) and vacuum tubular (VT) or atmospheric-vacuum tubular (AVT) units.

AT installations carry out shallow distillation of oil to produce fuel (gasoline, kerosene, diesel) fractions and fuel oil. VT units are designed for fuel oil distillation. The gas oil, oil fractions and tar obtained from them are used as raw materials for subsequent (secondary) processing processes to produce fuels, lubricating oils, coke, bitumen and other petroleum products.

Modern oil distillation processes are combined with the processes of dehydration and desalting, secondary distillation and stabilization of the gasoline fraction: ELOU-AT, ELOU-AVT, etc.

1. Description of the installation

The technological process in the atmospheric unit of the ELOU AVT-6 proceeds as follows. Oil dehydrated and desalted in ELOU is additionally heated in heat exchangers and fed for separation into partial topping column 1. The hydrocarbon gas and light gasoline leaving the top of this column are condensed and cooled in air and water cooling units and sent to an reflux tank. Part of the condensate is returned to the top of column 1 as acute reflux. The stripped oil from the bottom of column 1 is fed into a tubular furnace 4, where it is heated to the required temperature and sent to the atmospheric column 2. Part of the stripped oil from the furnace 4 is returned to the bottom of column 1 as a hot jet. Heavy gasoline is taken from the top of column 2, and fuel fractions 180-220 (230), 220 (230)-280 and 280-350 °C are removed from the side through stripping columns 3. The atmospheric column, in addition to acute irrigation, has two circulation irrigations, which remove heat below the fraction selection plates of 180-220 and 220-280 °C. Superheated water steam is supplied to the lower parts of the atmospheric and stripping columns to strip lightly boiling fractions. From the bottom of the atmospheric column, fuel oil is removed and sent to the vacuum distillation unit.

2. Technological diagram of the installation

In Fig. Figure 1 shows a schematic diagram of the atmospheric distillation unit of the AVT-6 ELOU installation.

1- topping column;

2 - atmospheric column;

3 - stripping columns;

4 - atmospheric furnace;

I - oil with ELOU;

II - light gasoline;

III - heavy gasoline;

IV - fraction 180-220;

V - fraction 220-280;

VI - fraction 280-350;

VII - fuel oil;

IX - water vapor.

3. Calculation of regulators

Table 1 Data for calculation

oil refining elow industry

To regulate the parameters, a three-circuit slave control system is used. The block diagram of such a system is shown in Fig. 2.

For a temperature control system in an atmospheric oven:

R1(s) - transfer function of the electric motor speed controller;

W11(s) - transfer function of the thyristor converter;

W12(s) - transfer function of the electric motor;

Woc1(s) - transfer function of the speed sensor;

R2(s) - transfer function of the fuel consumption regulator;

W21(s) - transfer function of the pump;

Woc2(s) - transfer function of the fuel consumption sensor;

R3(s) - transfer function of the temperature controller in an atmospheric furnace;

W31(s) - transfer function of the atmospheric furnace;

Woc3(s) is the transfer function of the atmospheric furnace temperature sensor.

Let's adjust the first circuit of the speed control system to the technical optimum (Fig. 3).

Desired transfer function of the first open loop:

On the other side:

By substituting the value into formula (2), we can calculate the controller transfer function:

Let's check the correctness of the calculations using computer simulation in Simulink. (Fig. 5) shows a graph of the transition process, the parameters of which correspond to the technical optimum.

Rice. 4 Diagram of the electric drive system model

Rice. 5 Transition graph

Transfer function of the first closed loop:

Let's set the second circuit of the fuel consumption control system to the technical optimum (Fig. 6).

Desired second open loop transfer function:

On the other side:

By substituting the value into formula (4), we can calculate the controller transfer function:

Let's check the correctness of the calculations using computer simulation in Simulink. (Fig. 8) shows a graph of the transition process, the parameters of which correspond to the technical optimum.

Rice. 7 Diagram of the electric drive system model

Rice. 8 Transition graph

Transfer function of the second closed loop:

Let's set the third circuit of the temperature control system to a symmetrical optimum (Fig. 9).

Desired third open loop transfer function:

On the other side:

By substituting the value into formula (6), we can calculate the controller transfer function:

Let's check the correctness of the calculations using computer simulation in Simulink. (Fig. 11) shows a graph of the transition process, the parameters of which correspond to the technical optimum.

Rice. 10 Diagram of the electric drive system model

Rice. 11 Transition graph

Conclusion

During this course work, regulators were calculated for each loop of the slave control system, the correctness of which was verified using computer simulation in Simulink. Based on the resulting graphs of the transient process, overshoot, mismatch time, maximum time and transient process time were calculated. The calculated values ​​correspond to the standard ones, depending on the selected condition (technical or symmetrical optimum). The technological process in the atmospheric unit ELOU AVT-6, which is characterized by high productivity, high flow rates and certain parameter values, the deviation of which is allowed only within the smallest limits, has also been studied in detail.

Posted on Allbest.ru

...

Similar documents

Tasks of the oil refining and petrochemical industry. Features of the development of the oil refining industry in the world. Chemical nature, composition and physical properties of oil and gas condensate. Industrial installations for primary oil refining.

course of lectures, added 10/31/2012


Importance of the chemical and petrochemical industry. Industry structure. Location of the chemical and petrochemical industries. Impact of the chemical and petrochemical industry on the environment. Current status and development trends.

abstract, added 10/27/2004


Types of industrial installations. Atmospheric oil distillation unit of the installation. Features of the technology of vacuum distillation of fuel oil using the oil version. Cross-flow landing columns for precise fractionation of fuel oil to produce oil distillates.

abstract, added 07/14/2008


Structure of the Moscow Oil Refinery in Kapotnya: 8 main and 9 auxiliary workshops, which include 48 process units. Data on the installation of ELOU-AVT-6. Technological diagram of the installation for triple oil evaporation ELOU-AVT.

practice report, added 07/19/2012


Automation of the chemical industry. Purpose and development of a detailed design for hydrocracking, catalyst regeneration and diesel fuel hydrodearomatization units. Modeling of an automatic control system. Selection of automation tools.

course work, added 08/16/2012


Elemental composition of oil and characteristics of oil products. Justification for the choice and description of the technological scheme of the atmospheric column. Calculation of distillation column K-1, K-2, tubular furnace, heat exchanger, condenser and refrigerator, pump selection.

course work, added 05/11/2015


Development of a functional and structural diagram of an automated control system for the process of atmospheric distillation of oil. Development of connections and connections. Software and mathematical support of the system. Calculation of the economic effect from the implementation of automated control systems.

thesis, added 08/11/2011


History of the enterprise JSOC Bashneft. Responsibilities of a master of instrumentation and automation equipment. Technological process of field oil preparation. Its regulation using primary sensors and actuators.

practice report, added 04/09/2012


Rectification of binary mixtures. Atmospheric oil distillation unit. Unit design and technological process. Monitoring and regulation of the oil/water interface level in an electric dehydrator. Development of a functional diagram of device automation.

course work, added 01/07/2015


The process of primary distillation of oil, its diagram, main stages, specific features. The main factors determining the yield and quality of products of primary distillation of oil. Installation with double evaporation of oil, yield of primary distillation products.

AUTOMATION OF TYPICAL TECHNOLOGICALPROCESSES

2.1. SEQUENCE OF SELECTION OF AUTOMATION SYSTEM*

General management task technological process is usually formulated as a problem of maximizing (minimizing) a certain criterion (cost, energy costs, profit) while fulfilling the restrictions on technological parameters imposed by regulations. Solving such a problem for the entire process as a whole is very labor-intensive, and sometimes almost impossible due to the large number of factors influencing the process. Therefore, the entire process is divided into separate sections. which are characterized by a relatively small number of re-

,*This chapter discusses the most characteristic features of regulation of basic technological parameters and processes. Based on the equation of the material and thermal balance of devices, they are analyzed as objects of regulation and a choice of options for control systems is given, starting with the simplest single-circuit ASR with gradual complication of circuits. In the sections devoted to the automation of reactors, heat exchangers and distillation columns, using the example of the simplest devices, the methodology for deriving linearized models of the statics and dynamics of technological objects with lumped and distributed parameters, which can be used in the calculation of control systems, is illustrated.

money. Typically, these sections coincide with completed technological stages, for which their own control subtasks can be formulated, subordinate to the general task of managing the process as a whole.

Control tasks for individual stages are usually aimed at optimizing (in a particular case, stabilizing) a process parameter or criterion that is easily calculated from measured operating parameters (productivity, product concentration, degree of conversion, energy consumption). The optimization of the criterion is carried out within the limits set by the technological regulations. Based on the problem of optimal control of individual stages of the process, they formulate automatic control tasks technological parameters for individual devices.

An important stage in the development of an automation system is analysis of main devices as objects of regulation, i.e. identification of all significant input and output variables and analysis of the static and dynamic characteristics of disturbance and regulation channels. The initial data in this case are a mathematical model of the process and (as a first approximation) a static model in the form of equations of material and heat balances. Based on these equations, taking into account the actual operating conditions of the apparatus, all significant factors influencing the process are divided into the following groups.

Disturbances that allow stabilization. These include independent technological parameters that may experience significant fluctuations, but according to operating conditions they can be stabilized using an automatic control system. Such parameters usually include some indicators of input flows. Thus, power consumption can be stabilized if there is a buffer capacity in front of the device, smoothing out fluctuations in the flow rate at the output of the previous device; stabilization of the supply temperature is possible if a heat exchanger is installed in front of the device, etc. Obviously, when designing a control system, it is advisable to provide for automatic stabilization of such disturbances. This will improve the quality of process management as a whole. In the simplest cases, based on such systems for automatic stabilization of disturbances, an open-loop (relative to the main indicator of the process) automation system is built, ensuring stable operation of the process within the framework of technological regulations.

Controlled disturbances. These conventionally include those disturbances that can be measured, but impossible or unacceptable to stabilize (power consumption supplied directly from the previous device; ambient temperature, etc.). The presence of significant unstabilized disturbances requires the use of either closed control systems according to the main process indicator, or

combined ASRs, in which the quality of regulation is improved by introducing dynamic disturbance compensation.

Uncontrollable disturbances. These include those disturbances that are impossible or impractical to measure directly. The first is a drop in catalyst activity, a change in heat and mass transfer coefficients, etc. An example of the second is the pressure of heating steam in a factory network, which fluctuates randomly and is a source of disturbance in thermal processes. Identifying possible uncontrolled disturbances is an important step in process research and control system development. The presence of such disturbances requires, as in the previous case, the mandatory use of automation systems that are closed according to the main indicator of the process.

Possible regulatory impacts. These are material or heat flows that can be changed automatically to maintain controlled parameters.

Output variables. From among them, adjustable coordinates are selected. When constructing closed control systems, technological parameters are chosen as controlled coordinates, the change of which indicates a violation of the material or thermal balance in the apparatus. These include: liquid level- liquid phase balance indicator; pressure- gas phase balance indicator; temperature- indicator of heat balance in the apparatus; con-"centration- material balance indicator for the component.

Analysis of possible regulatory influences and output coordinates of the object allows you to select control channels for the designed ASR. Moreover, in some cases the solution is determined unambiguously, while in others it is possible to select both the controlled coordinate and the control action for a given output. The final selection of control channels is carried out on the basis of a comparative analysis of the static and dynamic characteristics of various channels. In this case, such indicators as gain, pure delay time, its ratio to the largest channel time constant t are taken into account /T(see section 1.4).

Based on the analysis of the technological process as an object of regulation, an automation system is designed that provides a solution to the stated regulation problem. Start with designing single-circuit ACP for individual parametersmoat: they are the easiest to set up and reliable in operation, therefore they are widely used in the automation of process facilities.

However, with unfavorable dynamic characteristics of the control channels (large net delay, large t/G ratio), even in the case of optimal settings of the regulators, the quality of transient processes in single-circuit automatic control systems may turn out to be unsatisfactory. For such volumes

projects are analyzing the possibility construction of multi-circuitASR, in which the quality of regulation can be improved by complicating automation schemes, i.e. using cascade, combined, interconnected automated control systems.

The final decision on the use of one or another automation scheme is made after modeling of various automated systems And quality comparisons resulting regulation processes.

2.2. REGULATION OF BASIC TECHNOLOGICAL PARAMETERS

The main technological parameters that are subject to control and regulation in chemical technological processes include flow, level, pressure, temperature, pH value and quality indicators (concentration, density, viscosity, etc.)*. Flow regulation. The need to regulate flow occurs when automating almost any continuous process. Flow ACS, designed to stabilize disturbances in material flows, are an integral part of open-loop automation systems for technological processes. Flow ASRs are often used as internal circuits in cascade systems for regulating other parameters. To ensure a given mixture composition or to maintain material and thermal balances in the apparatus, systems are used to regulate the ratio of flow rates of several substances in single-circuit or cascade ASRs.

Flow control systems are characterized by two features: low inertia of the regulated object itself; the presence of high-frequency components in the flow change signal, caused by pressure pulsations in the pipeline (the latter are caused by the operation of pumps or compressors or random flow fluctuations when throttling the flow through a restriction device).

In Fig. 2.1 shows a schematic diagram of an object for regulating flow. Typically, such an object is a pipeline section between the flow measurement point (for example, the installation location of the restriction device 1) and the regulatory body 2. The length of this section is determined by the rules for installing restriction devices and regulators and is usually several meters. The dynamics of the channel “flow of substance through the valve - flow of substance through the flow meter” is approximately described by a first-order aperiodic link with pure delay. The pure delay time is usually

· The basics of measuring these parameters, automatic control devices and actuators are studied in the courses “Technological measurements and instruments” and “Technical automation equipment”. Here, the features of regulating these parameters are considered, taking into account the static and dynamic characteristics of control channels, control devices and automation equipment, and examples of the most common systems for regulating certain parameters are given.

sets fractions of seconds for gas and several seconds for liquid; the time constant is several seconds.

Due to the low inertia of the regulated object, special requirements are imposed on the choice of automation equipment and methods for calculating ACP. In particular, in industrial installations, the inertia of flow control and regulation circuits becomes commensurate with the inertia of the object, and it should be taken into account when calculating control systems.

An approximate estimate of the net delay and time constants of individual circuit elements shows (Fig. 2.2) that modern primary flow converters, built on the principle of dynamic compensation, can be considered as amplifiers. The actuator is approximated by a first-order aperiodic link, the time constant of which is several seconds, and the performance of the actuator increases significantly when using positioners. The impulse lines connecting the means of control and regulation are approximated by a first-order aperiodic link with pure delay, the parameters of which are determined by the length of the line and lie within a few seconds. For large distances between circuit elements, it is necessary to install additional power amplifiers along the length of the impulse line.

4 Due to the low inertia of the object, the operating frequency may be higher than the maximum, limiting the area of ​​normal operation of the industrial regulator, within which standard regulation laws are implemented. Outside this area, the dynamic characteristics of the controllers differ from the standard ones, which requires adjustments to the operating settings taking into account the actual control laws.

1 The choice of control laws is dictated by the usually required quality of transient processes. To regulate flow

Without static error, single-circuit ASRs use PI regulators. If the flow ACP is an internal loop in a cascade control system, re-

The flow regulator can implement the P-law of regulation. If there is high-frequency interference in the flow signal, the use of regulators with differential components in the control law without preliminary smoothing of the signal can lead to unstable operation of the system. Therefore, in industrial flow control systems, the use of PD or PID controllers is not recommended.

Flow control systems use one of three methods for changing flow:

throttling the flow of a substance through a regulatory body installed on the pipeline (valve, gate, gate);

changing the pressure in the pipeline using a controlled energy source (for example, changing the speed of the pump motor or the angle of rotation of the fan blades);

bypassing, i.e. transferring excess substance from the main pipeline to the bypass line.

The flow rate after the centrifugal pump is controlled by a control valve installed on the discharge pipeline (Fig. 2.3a). If a piston pump is used to pump liquid, the use of such an ACP is unacceptable, since during operation of the regulator the valve may close completely, which will lead to a rupture of the pipeline (or to surging if the valve is installed on the pump suction). In this case, flow bypass is used to regulate the flow (Fig. 2.3.6).

Regulation of the flow of bulk substances is carried out by changing the degree of opening of the control valve at the outlet of the hopper (Fig. 2.4, a) or by changing the speed of the conveyor belt (Fig. 2.4,6). In this case, the flow meter can be a weighing device that determines the mass of material on the conveyor belt.

Cost ratio adjustment two substances can be carried out according to one of three schemes described below.

1. With an unspecified overall productivity, the consumption of one substance (Fig. 2.5, a) G1, called “leading”, can change arbitrarily; the second substance is supplied at a constant ratio y s first, so the “slave” flow rate is yg1.

Rice. 2.4. Schemes for regulating the consumption of bulk solids:

A- changing the degree of opening of the control valve; b - changing the speed of the moving conveyor, 1 - bunker; 2 - conveyor; 3 - regulator; 4 - regulating valve; 5 - electric motor

Sometimes a ratio relay is used instead of a ratio regulator.wearing and a conventional regulator for one variable(Fig. 2.5,b). Relay output 6, establishing a given ratio coefficienty, submitted in the form of a regulation tasklator 5, ensuring the maintenance of the “driven” flow rate.

2. At a given “leading” flow rate, in addition to the ASR ratio ASR of the “leading” flow rate is also used (Fig. 2.5, c). When that which scheme in case of changing the flow targetg1 automatic consumption will also change significantlygz (in a given ratio withg1).

3. ACP cost ratio is an internal cont. rum in the cascade control system of the third technological process sky parameter at(for example, the temperature in the device). At

Rice. 2.5. Schemes for regulating the cost ratio:

a, b- with an unspecified total load, V- for a given total load, g - for a given total load and correction of the ratio coefficient according to the third parameter; /, 2 - flow meters, 3 - ratio regulator; 4, 7 - control valves; 5 - flow regulator, 6 - ratio relay, 8 - temperature regulator; 9 - limiting device

In this case, the given ratio coefficient is set by an external controller depending on this parameter so that G 2 = y(y) G1 (Fig. 2.5, d). As noted above, the peculiarity of setting up cascade ACPs is that the xph limit is set on the internal controller’s task [hrn,xpB], then the setting for the ratio regulator remains at the maximum permissible value y (i.e. yn or yb) - Level control. The level is an indirect indicator of hydrodynamic equilibrium in the apparatus. Constancy of the level indicates compliance with material balance, when the influx of liquid is equal to the flow, and the rate of change of level is zero. It should be noted that “inflow” and “sink” are general concepts here. In the simplest case, when no phase transformations occur in the apparatus (collectors, intermediate tanks, liquid-phase reactors), the inflow is equal to the flow rate of liquid supplied to the apparatus, and the drain is equal to the flow rate of liquid removed from the apparatus. In more complex processes, accompanied by a change in the phase state of substances, the level is a characteristic of not only hydraulic, but also thermal and mass transfer processes, and the influx and drainage take into account the phase transformations of substances. Such processes take place in evaporators, condensers, evaporation units, distillation columns, etc.

In the general case, the change in level is described by an equation of the form

where S is the area of ​​the horizontal (free) section of the apparatus; Ginx, Gout - liquid flow rates at the entrance to the apparatus and exit from it; Gvol - the amount of liquid produced (or consumed) in the apparatus per unit of time.

Depending on the required accuracy of level maintenance, one of the following two control methods is used:

1) positional control, in which the level in the apparatus is maintained within specified, fairly wide limits;

lah: lh

Rice. 2.6. Example of a positional level control circuit:

1 - pump; 2 - apparatus, 3 - level indicator; 4 - level regulatornya, 5, 6 - control valves

Figure 2 7. Continuous level control circuits:

A - regulation "on the inflow"; b - regulation “at the drain”, V- cascade ASR; 1~level regulator, 2 - control valve; 3, 4 - flow meters, 5 - ratio regulator

(Fig. 2.6). When the level limit is reached, the flow is automatically switched to a spare tank;

2) continuous regulation, which ensures stabilization of the level at a given value, i.e. L = L°.

Particularly high demands are placed on the accuracy of level control in heat exchangers, in which the liquid level significantly affects thermal processes. For example, in steam heat exchangers, the level of condensate determines the actual heat exchange surface. In such ASRs, PI controllers are used to regulate the level without static error. P-regulators are used only in cases where high quality control is not required and disturbances in the system do not have a constant component, which can lead to the accumulation of a static error.

In the absence of phase transformations in the apparatus, the level in it is adjusted in one of three ways:

changing the liquid flow rate at the entrance to the apparatus (“inflow” regulation, Fig. 2.7, a);

changing the liquid flow rate at the outlet of the apparatus (control “at the drain”, Fig. 2.7.6);

regulation of the ratio of liquid flow rates at the entrance to the apparatus and exit from it with level correction (cascade ASR, Fig. 2.7,c); disabling the correction circuit can lead to the accumulation of errors when regulating the level, since due to inevitable errors in setting the ratio regulator, the liquid flow rates at the inlet and outlet of the device will not be exactly equal to each other and due to the integrating properties of the object [see. equation (2.1)] the level in the apparatus will continuously increase (or decrease).

In the case when hydrodynamic processes in the apparatus are accompanied by phase transformations, the level can be adjusted by changing the supply of coolant (or coolant), as shown in Fig. 2.8. In such devices, the level is interconnected with other parameters (for example, pressure), so the choice of a method for regulating the level in each specific

Rice. 2.8. Level control circuit in the evaporator:

1 - evaporator; 2 - level regulator; 3 - control valve

Rice. 2.9. Fluidized bed level adjustment:

A- removal of granular material; b - change in gas consumption; 1 - fluidized bed apparatus; 2 - level regulator; 3 - regulatory body

In this case, it must be carried out taking into account the remaining control loops.

A special place in level control systems is occupied by level ASRs in devices with a fluidized (fluidized) bed of granular material. Stable maintenance of the fluidized bed level is possible within a fairly narrow range of the ratio of gas flow and bed mass. With significant fluctuations in gas flow (or flow of granular material), a regime of layer entrainment or sedimentation occurs. Therefore, particularly high demands are placed on the accuracy of fluidized bed level control. The flow of granular material at the inlet or outlet of the apparatus (Fig. 2.9, a) or the gas flow for liquefying the layer (Fig. 2.9,6) are used as regulatory influences. Pressure regulation. Pressure is an indicator of the ratio of gas phase flow rates at the entrance to and exit from the apparatus. Constancy of pressure indicates compliance with the material balance in the gas phase. Typically, the pressure (or vacuum) in a process installation is stabilized in any one device, and throughout the system it is established in accordance with the hydraulic resistance of the line and devices. For example, in a multi-effect evaporator installation (Fig. 2.10), the vacuum in the last evaporator is stabilized. In other devices, in the absence of disturbances, a vacuum is established, which is determined from the conditions of material and thermal balances, taking into account the hydraulic resistance of the production line.

In cases where pressure significantly affects the kinetics of the process, a pressure stabilization system is provided in individual devices. An example is the rectification process, for which the phase equilibrium curve depends significantly on pressure. In addition, when regulating the binary distillation process, often as an indirect

indicator of the composition of a mixture, its boiling point is used, which is uniquely related to the composition only at constant pressure. Therefore, in product distillation columnsusually provide special stabilization systemspressure (Fig. 2.11).

Equation of material balance of the apparatus in the gas phaseis written in the form:

WhereV- volume of the apparatus;GinputAndGOut- gas flow rate respectively supplied to the apparatus and removed from it; C0b is the mass of gas produced (or consumed) in the apparatus per unit time.

As can be seen from a comparison of equations (2.1) and (2.2), methods of repressure regulation is similar to regulation methods level. In the examples of pressure control systems discussed above, the regulating influences selected the non-condensing flow rate exhaust gases removed from the top of the column (i.e.GOUT, rice. 2.11) and cooling water flow into the barometric conea densator that affects the rate of condensation of the secondarypair (i.e. onG0b, rice. 2.10).

Regulatory systems occupy a special place among pressure control systems.determining the pressure drop in the apparatus, which characterizeshydrodynamic regime, which significantly affects the performanceflow of the process. Examples of such devices includepacked columns (Fig. 2.12, a), fluidized bed apparatus(Fig. 2.12.6), etc.

Temperature regulation. Temperature is an indicatorthermodynamic state of the system and is used as a

Rice. 2.10. Vacuum control in a multi-effect evaporator mouthnew:

/, 2 - evaporators; 3 - barometric capacitor; 4 - vacuum regulator; 5 - control valve

Rice. 2.11. ASR pressure in the distillation column:

1 - column; 2 - reflux condenser; 3 - reflux tank; 4 - pressure regulator; 5 - control valve

Rice. 2.12. Differential pressure control circuit:

a - in a column apparatus with a nozzle;b - in a fluidized bed apparatus; 1 - apparatus; 2 - differential pressure regulator; 3 - control valve

moving coordinate when regulating thermal processes. Dynamic characteristics of objects in temperature control systems

depend on the physical and chemical parameters of the process and the design of the apparatus. Therefore, it is impossible to formulate general recommendations for choosing temperature ACP, and an analysis of each specific process is required.

General features of temperature control systems include significant inertia of thermal processes and industrial temperature sensors. Therefore, one of the main tasks when designing temperature control systems is to reduce the inertia of sensors.

Let us consider, for example, the dynamic characteristics of thermal
torque meter in a protective case (Fig. 2.13, a). Block diagram ter
a torque meter can be thought of as a series connection
four thermal containers (Fig. 2.13.6): protective cover /,
air gap 2, thermometer walls 3 and actually ra
barrel of liquid 4. If we neglect thermal resistance
each layer, then all elements can be approximated
periodic links of the 1st order, the equations of which have
This is the view:

mi - the mass of the cover, air gap, wall and liquid, respectively; Cpi- specific heat capacities; a j1, a j2 - heat transfer coefficients; F f1 , F f2 - heat transfer surfaces.

As can be seen from equations (2.3), the main directions for reducing the inertia of temperature sensors are:

increasing heat transfer coefficients from the medium to the cover inas a result of the correct choice of sensor installation location; atIn this case, the speed of movement of the medium should be maximum; atother things being equal, it is more preferable to install termometers in the liquid phase (compared to the gaseous phase), in concondensing steam (compared to condensate), etc.;

reduction of thermal resistance and thermal capacityprotective cover as a result of the choice of its material and thicknessshins;

decrease in the time constant of the air gap fordue to the use of fillers (liquid, metalshavings); for thermoelectric converters (thermocouples)the working junction is soldered to the protective cover;

selection of the type of primary converter; for example, when choosing a resistance thermometer, thermocouple or manometric thermometer, it is necessary to take into account that the low-inertia thermocouple has the least inertia, and the manometric thermometer has the greatest. pH regulation. pH control systems can be subdivideddivided into two types, depending on the required accuracy of regulationlation. If the rate of change in pH is small, and letthe possible limits of its fluctuations are quite wide; they are used according topositional control systems that maintain pH levelsgiven limits: рНi<рН<рНв. Ко второму типу относятся systems that provide regulation of processes in whichrequires precise maintenancepHat a given value (atexample, in neutralization processes). To regulate them, useContinuous PI or PID controllers are used.

A common feature of objects when regulating pH isthere is a nonlinearity of their static characteristics associated withnonlinear dependence of pH on reagent consumption.In Fig. Figure 2.14 shows a titration curve characterizing the

Rice. 2.14. Dependence of pH value onreagent consumption

dependence pHsour from consumptionYoug1 . For various givenpH values ​​on this curve can beidentify three characteristic areas:first (middle), related to

almost neutral media, is close to linear and characterizes has a very high gain; the second and third sections, related to highly alkaline or acidic environments, have the greatest curvature.

In the first section, the object in its static characteristics approaches the relay element. In practice, this means that when calculating a linear ACP, the gain of the regulator is so small that it goes beyond the operating settings of industrial regulators. Since the neutralization reaction itself takes place almost instantly, the dynamic characteristics of the devices are determined by the mixing process and in devices with mixing devices they are quite accurately described by differential equations of the 1st order with a delay. Moreover, the smaller the time constant of the device, the more difficult it is to ensure stable control of the process, since the inertia of the devices and the controller and the delay in the impulse lines begin to affect.

To ensure stable pH regulation, special systems are used. In Fig. 2.15, and shows an example of a system pH regulation with two control valves. Clapan 1, having a large nominal diameter, serves forrough flow control and set to the maximum range of change in the controller output signal [X rn , hrv](Fig. 2.15.6, curve 1). Valve 2, serving for precise regulation, designed for lower throughput and configured in such a way that whenxp = xp °+ A he's completely offcovered, and Хр=хр°-A - completely closed (curve 2). So

Rice. 2.15. Example of a pH control system:

a - functional diagram; b - static characteristics of valves; /, 2 - control valve; 3 - pH regulator

Rice. 2.16. Piecewise linear approximation of the static characteristics of an object when regulating pH

Rice. 2.17. Block diagram of a pH control system with two regulators

Thus, with a slight deviation of pH from pH°, when xp°-A 2. If \xp-xр0|>|D, valve 2 remains in the extreme position, and regulation is carried out by valve /.

In the second and third sections of the static characteristic (Fig. 2.14), its linear approximation is valid only in a very narrow range of pH changes, and in real conditions the control error due to linearization may turn out to be unacceptably large. In this case, more accurate results are obtained by piecewise linear approximation (Fig. 2.16), in which the linearized object has a variable gain:

In Fig. Figure 2.17 shows a block diagram of such an ASR. Depending on the mismatch A pH, one of the regulators is switched on, adjusted to the corresponding gain of the object.

Transcript

1 Ministry of General and Professional Education of the Russian Federation Tver State Technical University V.F. Commissioner Automatic control of technological processes Textbook Tver

2 UDC 6.5 Automatic control of technological processes: Textbook Second edition, expanded / V.F. Commissioner; Tver State Technical University, Tver, 48с. Methods for calculating automatic control systems for technological processes of various types are considered. Intended for specialty students. “Automation of technological processes and production” when they studied the discipline of the same name. Prepared at the Department of Automation of Technological Processes of Tver State Technical University.

3 3 Introduction One of the most important tasks of automation of technological processes is automatic control, which aims to maintain constancy, stabilize the set value of controlled variables or change them according to a law specified in time; program control with the required accuracy, which allows us to ensure the production of products of the required quality, as well as safe and economical operation of technological equipment. The controllable variables are usually the operating level, temperature, pressure, flow rate or qualitative humidity, density, viscosity, composition, etc. indicators of the functioning of technological processes, characterizing the material or energy balance in devices and the properties of the product. The task of automatic regulation is implemented through automatic control systems ACP. The block diagram of a closed ASR is shown in Fig.. F RO x OP S P - back Fig..

4 4 In Fig. designated: OR object of regulation, technological process or apparatus; y controlled variable; x regulatory influence by which the regulatory process is carried out. Regulating influences are usually the flow rates of liquid, gaseous, and granular bodies; RO is a regulating working body with the help of which the energy consumption of a substance is changed. To change the flow rates of liquid and gaseous bodies, throttling-type working bodies with a variable flow area are widely used; S is the position of the working element, usually measured in % of stroke PO, for example, the movement of a valve stem or the rotation of a damper. Since the regulatory impact x, as a rule, is not measured, S is usually taken as the regulatory impact, thereby classifying RO as the object of regulation; F - disturbing influences that influence the value of the controlled variable; R - automatic regulator - a set of elements designed to solve the regulation problem; set - the set value of the controlled variable, which must be maintained by the controller; - a comparing device that generates an error mismatch signal: back As an example in Fig. shows a diagram for regulating the temperature of the product θ pr at the outlet of the heat exchanger by changing the coolant supply G.

5 5 G pr θ pr R G Fig.. One of the main disturbances in this system is the flow rate of the heated product G pr. The reason for regulation in a closed ASR is the occurrence of an error. When it appears, the controller changes the regulatory action x until the error is completely eliminated in an ideal system. Thus, the ACP is designed to maintain the controlled variable at a given level when disturbances fluctuate within certain limits. In other words, the main task of the regulator is to eliminate the discrepancy by changing the regulatory impact. The most important advantage of a closed-loop automatic control system is that it reacts to any disturbance that leads to a mismatch. At the same time, such systems are fundamentally characterized by a control error, since the occurrence

6 6 mismatch always precedes its elimination and, in addition, a closed ASR under certain conditions can become unstable. The main tasks that arise when calculating ACP are:. Mathematical description of the regulated object;. Justification of the structural diagram of the automated control system, the type of regulator and the formation of requirements for the quality of regulation; 3. Calculation of controller settings; 4. Analysis of the quality of regulation in the system. The purpose of calculating a closed-loop automatic control system is to ensure the required quality of regulation. By quality of regulation we will understand the values ​​of indicators that characterize the shape of the curve of the transient process in a closed automatic control system with a stepwise effect at its input. An approximate view of the transient characteristics of a closed-loop automatic control system along the channels of the master and disturbing regulatory influences in a particular case is shown in Fig. 3. Transient response of a closed-loop system along the reference channel, line y fact in Fig. 3a reflects the nature of the transition of the controlled variable from one steady value to another. x a y back b y id y fact y fact y id Fig. 3.

7 7 It would be ideal if this transition occurred abruptly line y id Transition characteristic along the channel of regulatory influence line y fact in Fig. 3b reflects the process of suppression of disturbance by the system. The ideal would be for the system not to react at all to disturbances in the ID line. This manual discusses methods for solving typical problems that arise when calculating automated control systems of various types that are used in the practice of automation of technological processes.. Mathematical description of regulated objects [4].. Main characteristics and properties of regulated objects A regulated object can be in one of two states: statics or dynamics. Statics is a steady state in which the input and output quantities of an object are constant in time. This definition is valid for stable static objects. Dynamics is a change in time of the output variable of an object due to a change in the input variable or non-zero initial conditions. Static characteristics of regulated objects The behavior of a regulated object in statics is characterized by the static “input-output” characteristic, which represents the dependence between the steady-state values ​​of the output and input variables: f st st By the type of static characteristics, linear and nonlinear objects are distinguished. The static characteristic of a linear object represents a straight line passing through the origin of coordinates with the equation

8 8 K The characteristic with the equation K b, which does not pass through the origin of coordinates, can be reduced to linear, denoting b ". Objects whose static characteristics differ from the straight line are nonlinear. The slope angle of the static characteristic α, equal to the derivative of the output variable with respect to the input, is called the static transfer coefficient of the object: K lim gα The coefficient K has the dimension: units of the output variable per unit of input influence. Physical meaning: the change in the controlled variable per unit of input influence, i.e. the transfer coefficient characterizes the slope of the static characteristic. For linear objects. Ku/ constant, for nonlinear K is When calculating the ACP, nonlinear characteristics are usually linearized. The linearization of the tangent by a linear approximation of the Taylor series expansion is widely used. take into account that the accuracy of linearization decreases with increasing increment value, therefore linearization of the tangent is valid only in

9 9 a sufficiently small neighborhood of the point x. In addition, since the expression includes the derivative of the function f, this linearization method is only suitable for differentiable functions. Dynamic characteristics of regulated objects. Differential equation The main dynamic characteristic of the objects of regulation is the differential equation. Objects can be described by two types of differential equations: ordinary differential equations and partial differential equations. Ordinary differential equations describe objects with lumped parameters, which can conventionally be considered containers with ideal instantaneous mixing. Variables in such objects depend only on time and do not depend on the coordinates of the point of measurement of the variable. Partial differential equations describe objects with distributed physical parameters, these are usually devices in which one of the coordinates is much larger than the others, for example, a “pipe-in-pipe” heat exchanger, column-type devices, etc. In such objects, the values ​​of the variables depend not only on time, but also the coordinates of the point of measurement of the variables, therefore, differential equations include not only derivatives with respect to time, but also with respect to coordinates. Typically, in calculations, partial differential equations are approximated by a system of ordinary differential equations. In the future we will consider objects described by ordinary differential equations of the form: d d n n n n< n n n d d m d d L bm L b ; m, m d d

10 where n is the order of the left side and the entire equation as a whole, m is the order of the right side. Since real control objects represent inertial links, always m

11 Basic properties of the Laplace transform. The delay of the argument by τ corresponds to the multiplication of the image by τ e the original displacement theorem, i.e. L e τ ( τ) 4 This property allows you to find images of differential equations with a retarded argument. Differentiation of the original under zero initial conditions corresponds to multiplying the image by p: d L d, therefore formally the variable p can be considered a symbol of differentiation. In static p. In the general case d L d 5 Since integration is the inverse action of differentiation, the integration of the original corresponds to dividing the image by p: ( d) L / Property 5 allows us to write the Laplace image of the differential equation: n n n n m L bm L b Thus, the Laplace image of the differential equation represents an algebraic expression that can be resolved relative to the image of the output variable ur, and then again move from the image to the original. This operation is called the inverse Laplace transform and is denoted by the operator L ( ) L:

12 The inverse Laplace transform is determined by the integral α j π e d j α j To facilitate finding an image from the original and the original from the image, correspondence tables have been compiled between the originals and their images for the simplest functions. These tables are given in manuals on the Laplace transform and in textbooks on control theory. To find the originals of complex images, use the formula for decomposing the image into simple fractions. cm n The ratio of the Laplace image of the output variable to the image of the input variable under zero initial conditions is called the transfer function W bm n m n L b L, in the form: or, since b, the transfer function can be written in b W L L m m n n B, A where Ap and Bp are polynomials from p to orders n and m, respectively. What is the relationship between the transfer function and the static transfer coefficient? The transfer function is a dynamic characteristic, the transfer coefficient is a static characteristic. Statics at rest is a special case of dynamics of motion. Consequently, K is a special case of W in statics. Since in statics p, then K W 6

13 3 Temporal characteristics The temporal characteristic of an object is its response to a typical aperiodic signal. The step function or its derivative, the δ function, is most often used as input signals. The response of an object or any dynamic link to a step function of unit amplitude (unit step function) is called the transient response of the object of link h. The reaction of an object to a step of arbitrary amplitude x is called the acceleration curve of the object, Fig. 4. To obtain the transient response from the acceleration curve y, each ordinate of the acceleration curve should be divided by the step amplitude: h / Fig. 4. Fig. 5. The response of an object to the δ function in real conditions to a pulse of finite duration and amplitude, for example, a rectangular one, is called the impulse response, the weight function of the control object, Fig. 5.

14 4 Frequency characteristics Determine the behavior of an object in the frequency domain when a harmonic signal is applied to its input: m sin, where πf π / - circular frequency of the signal, f - frequency, - signal repetition period, x m amplitude of the signal. At the output of the linear object, harmonic oscillations of the same frequency also appear, but with a different amplitude and phase (Fig. 6: ϕ m ϕ; 36, j m m ϕ j Fig. 6. Fig. 7. The values ​​of m and ϕ depend on the frequency of the input signal. Since we are interested in changing two quantities of amplitude and phase at once, it is convenient to consider the frequency characteristics in the complex plane. The harmonic input signal is represented on the complex plane by a vector j, the modulus of which is equal to the amplitude x m, and the inclination angle argument is equal to the oscillation phase (Fig. 7: j m e j The symbol in this case means “depicted.”

15 5 Similarly, the output signal of the object is depicted in the complex plane by vector j: m e j ϕ j Images j and j are called Fourier images, Fourier spectra of harmonic signals and. The ratio of the Fourier images of the output harmonic signal to the input is called the frequency transfer function FFT or complex frequency response W j: j m jϕ W j e j m A e jϕ The module of the frequency transfer function A at frequency determines the transmission coefficient of the object at a given frequency, ϕ is the phase shift between the output and input signals at frequency. The transfer function is a function of the complex variable α j. The frequency transfer function is a function of the imaginary variable j. Therefore, the frequency transfer function is a special case of the transfer function when the variable p takes on a purely imaginary value j. Therefore, a formal expression for the frequency transfer function can be found by replacing the variable p in the transfer function W with j, i.e. assuming j: bm W j j n m j n LL b LL What is the difference between a transfer function and a frequency transfer function? The transfer function reflects the behavior of the control object or any dynamic link in dynamics under an arbitrary form of input action. The frequency transfer function reflects

16 6 behavior of the link object only in the steady state of harmonic oscillations. Thus, the frequency transfer function is a special case of the transfer function in the same way as an imaginary variable is a special case of the complex variable p. j is The frequency transfer function is written in algebraic form in Cartesian coordinates: W j P jq, [ W j ]; Q Jm[ W j ], P Re or in exponential form in polar coordinates: W j W j A e jϕ [ W j ] A W j; ϕ rg Hodograph of the vector W j the graph described by the end of the vector as the frequency changes from o to is called the amplitude-phase characteristic of the AFC. AFC shows how the amplitude ratios and the phase shift between the output and input signals change when the frequency of the input signal changes (Fig. 8. Dependences of the ratio of the amplitudes of the output and input signals A and the phase shift between the output and input signals ϕ on frequency are called amplitude-frequency response and phase-frequency response characteristics, respectively, Fig. 9. The AFC contains the same information about the object link as the AFC and PFC combined. j A ϕ ϕ A Fig. 8. Fig. 9.

17 7 Basic properties of regulated objects. Load Load is the amount of substance or energy taken away from the regulated object during operation. Load changes, as a rule, are the main disturbing influence in the control system, because leads to an imbalance between the inflow and outflow of energy matter in the object, which causes a change in the controlled variable, for example, the level of liquid in the container (Fig. Q pr H Q st Fig.. In addition, a change in load leads to a change in the dynamic characteristics of the object. For example, in a container with perfect mixing of rice. the time constant is equal to the ratio of the volume of liquid stored in the container to the load, i.e. the time constant of this object is inversely proportional to the load. Capacity Capacity is the amount of energy substance that an object can accumulate. Capacity characterizes the inertia of the regulated object. Objects of regulation can be single- or multi-capacitive. Multi-capacity objects consist of two or more containers separated

18 8 transition resistances. The number of containers determines the order of the object's differential equation. For example, a container with liquid in Fig. refers to the number of single-capacity objects. An example of a three-capacity object is the shell-and-tube heat exchanger in Fig., in which the heated liquid receives heat through the walls of the tubes from the coolant. The first container is the amount of heat in the heated liquid in the interpipe space. The second container is the amount of heat in the coolant inside the tubes. The third capacity is the amount of heat in the walls of the pipes; this capacity is usually small compared to the others, and is neglected. Self-leveling Self-leveling is the ability of an object to restore balance between the inflow and outflow of a substance of energy due to a change in the controlled variable due to internal negative feedback in the controlled object. For example, in a container with free drainage of rice. as the inflow increases, the level increases and, due to this, the runoff increases until the balance between the inflow and runoff is restored. The greater the self-leveling value, the less the controlled variable deviates under the influence of disturbances. Thus, self-leveling facilitates the operation of the automatic regulator. Depending on the magnitude of self-leveling, control objects can be divided into objects with positive, zero and negative self-leveling. From a dynamic point of view, objects with positive self-alignment are stable inertial links. Their transient characteristics end at steady state

19 9 the section where the controlled variable comes to rest and stops changing Fig., curve. 3 Fig.. Quantitatively, the value of self-leveling is characterized by the self-leveling coefficient ρ, which represents the modulus of the reciprocal of the static transfer coefficient of the object: ρ K The self-leveling coefficient shows how much the input variable of the object must change in order for the output to change by one. Linear objects have constant self-leveling ρ cons, nonlinear objects have variable ρ Vr. Objects that do not have self-leveling and objects with zero self-leveling include the so-called neutral or astatic objects, which represent integrating links from a dynamic point of view. Changes in the controlled variable in such objects can be arbitrarily large. An example of neutral

20 of the object is a container with forced draining Fig. Here, at Qpr Qst, the level rises until the container overflows or drops to zero. Q pr N Q st Fig.. If there is equality between inflow and drainage, such an object can be in equilibrium at any value of the controlled variable, which is why it is called neutral or astatic. The steady-state section of the transition characteristic of an astatic object represents a straight line on which the controlled variable changes at a constant speed, the curve in Fig.. The equation of the ideal integrating link K d, whence d / d K The parameter K a, characterizing objects with zero self-leveling, is called the reduced acceleration rate of a neutral object and has the meaning of the rate of change of the controlled variable per unit of input influence. There are objects in which, under certain conditions, an uncontrollable process occurs. In these objects, the rate of change of the controlled variable in the transient process tends to

21 self-increasing curve 3 in Fig. Such objects are called objects with negative self-alignment. From a dynamic point of view, they are unstable links. For neutral and unstable objects ρ. Delay Delay is the time interval from the moment the disturbance is applied to the beginning of the change in the controlled variable. A distinction is made between pure and capacitive delay. Pure transport delay τ is the time that a flow of energy substance spends on traveling the distance from the point of application of the disturbance to the point of measurement of the controlled variable in a single-capacitive object. An example of a link with a pure delay is the conveyor belt feeder Fig. 3. The pure delay time is equal to the ratio of the length of the active section of the conveyor belt l to the linear speed of the belt V: τ l V Q n n V l Q П τ l nm Fig. 3. Fig. 4.

22 In multi-capacitive objects, several containers are connected in series, which causes a slowdown in the flow of energy substances from one container to another and leads to the occurrence of capacitive lag. Figure 4 shows the transient characteristics of one-n, two-n, and multi-capacitive nm objects. When the number of capacitances is n>, an inflection point P appears in the transient characteristic. As n increases, the initial section of the transient characteristic gravitates more and more towards the abscissa axis, as a result of which a capacitive delay τ e is formed. There is a fundamental difference between pure and capacitive lag. With pure lag, the controlled variable is zero throughout the entire lag time. With capacitive lag it changes, although very little. In the time domain, transport and capacitive delays appear approximately equally, but in the frequency domain, the behavior of these links differs significantly. Real objects usually contain both types of delay, as a result of which the total delay τ is equal to their sum: τ τ τ e It is almost impossible to separate the capacitive delay from the pure delay in the experimental characteristic. Therefore, if the net delay is determined from the experimental acceleration curve, its value is always subjective, i.e. depends on the researcher. The delay sharply worsens the quality of regulation in automated control systems... Methods for the mathematical description of regulated objects Methods for the mathematical description of regulated objects can be divided into analytical, i.e. no experiment required

23 3 at an industrial facility and experimental i.e. based on the results of the experiment. Analytical methods are called methods for obtaining mathematical models of objects based on the analysis of physical and chemical processes occurring in the object, taking into account its design and the characteristics of the processed substances. Advantages of analytical models of objects. No on-site industrial experiments are required. Therefore, these methods are suitable for finding models of objects at the stage of their design or when it is impossible to experimentally study the characteristics of regulated objects. Analytical models include the design characteristics of objects and indicators of the technological mode of their operation. Therefore, such models can be used to select the optimal design of the apparatus and optimize its technological regime. 3. Analytical models can be used for such objects. At the same time, analytical models are quite complex. In real objects, processes of three types can simultaneously occur: chemical transformations, heat and mass transfer. Simultaneous accounting of all these processes is quite a difficult task. Experimental methods for obtaining models include obtaining time or frequency characteristics as a result of an industrial experiment and their approximation, i.e. selection of an analytical relationship that describes the experimental data with the required accuracy. When taking time characteristics, the object is in a transition mode from one steady state to another. When taking frequency characteristics, the object is introduced into a steady-state mode of harmonic oscillations. Therefore, obtaining frequency

24 4 characteristics, in principle, allows one to obtain more representative information about an object, which is much less dependent on random disturbances acting on the object. But an experiment to take frequency characteristics is more labor-intensive than an experiment to take time characteristics and requires special equipment. Therefore, the most accessible in real conditions is to obtain time characteristics. It should be noted, however, that experimental models of objects can only be used for those objects and those conditions of their functioning for which the experiment was carried out..3. Obtaining and approximating the temporal characteristics of regulated objects. Preparation and conduct of an experiment When developing an experimental design for taking the temporal characteristics of regulated objects, issues related to the measurement and registration of the test effect and the controlled variable are resolved. Planning an experiment comes down to choosing the type of test effect, the magnitude of its amplitude and the number of experiments. To obtain the acceleration curve, a step function is used as a test effect. If a step effect is unacceptable for a controlled object without self-leveling or a long-term deviation of the controlled variable from the nominal value is unacceptable, a rectangular pulse type effect is used. The impulse transient response obtained in this way, in accordance with the principle of superposition for linear objects, can be rearranged into an acceleration curve.

25 5 When choosing the amplitude of the test effect, a compromise is sought between the following conflicting requirements. On the one hand, the amplitude of the input influence must be large enough to reliably isolate the useful signal from the background of measurement noise. On the other hand, too large deviations of the controlled variable can lead to disruptions in the operation of the facility, leading to a decrease in product quality or the occurrence of an emergency mode. In addition, with large disturbances, the nonlinearity of the static characteristics of the object is affected. When determining the number of experiments, it is useful to take into account the following factors: the linearity of the static characteristics of the object, the degree of noise in the characteristics, the magnitude of load fluctuations, and the non-stationary characteristics over time. Before conducting an experiment, the object must be stabilized in the vicinity of its nominal operating mode. The experiment to take the time characteristic continues until a new value of the controlled variable is established. When the object is noisy, the experimental characteristics are smoothed over time with high-frequency noise or over time with low-frequency noise. Approximation of transient characteristics of regulated objects. The approximation task includes three stages. Selection of the approximating transfer function. The transient characteristics of objects with self-leveling and lumped parameters are approximated by a fractional-rational transfer function in the general case with a pure delay of the form:

26 6 W about To about b m n m n LL e LL For objects without self-leveling in the denominator of the transfer function 7, the Laplace transform variable p is a sign of the integrating link is added as a factor. As practice shows, satisfactory approximation accuracy is achieved when using models for which n.3, and n-m in the absence of an inflection point in the acceleration curve and n-m in its presence.. Determination of the coefficients of the approximating transfer function. See below 3. Evaluation of approximation accuracy. To assess the approximation accuracy, it is necessary to construct a calculated characteristic and determine the maximum approximation error. Expressions for transient characteristics corresponding to some approximating transfer functions are given in table. When calculating on a computer in expressions for transient characteristics, one should go to discrete time τ 7 i sampling interval, and if there is a pure delay in model 7, the argument for i i for i > τ k Approximation of the transient characteristics of objects with self-leveling by a first-order inertial link with delay a Graphical method tangent method The transfer function is sought in the form:

27 7 W K e τ 8 To determine τ and T, a tangent AB is drawn to the transition characteristic in Fig. 5 at the inflection point C. The inflection point corresponds to the maximum angle α between the tangent and the abscissa axis of the mouth B C mouth O τ α A D The segment OA cut off by the tangent on the abscissa axis, is taken as the time of pure delay τ: τ OA The length of the subtangent projection of the segment AB on the abscissa axis is taken as T: TAD Fig. 5. The transfer coefficient K is found as the ratio of the increments of the output and input quantities in steady state: set K 9 set

28 8 Table. models Transfer function Roots of the characteristic equation Transient characteristic K e K, is the amplitude of the step action K α β e e K β α β α β α β 3 K α j ±, α α α rcg e K sin 4 b K α β e b e b K β α α β β α β α α β 5 b K α j ±, sin α α α α α b rcg e b b K α β γ 3 e e e K γ β α γ β γ α γ αβ γ β α β αγ γ α β α βγ K α j ±, γ 3 e rcg e γ α γ α α γ α α α γ γ α α γ sin 3 3 b K α β γ 3 e b e b e b K γ β α β γ α γ γ αβ γ β α β β αγ γ α β α α βγ

29 9 3 3 b К α j ±, γ 3 [ e b b b rcg e b b К γ α γ α γ α α γ α γ α α α α γ γ α α α γ sin

30 b Interpolation method The acceleration curve is preliminarily normalized from to using the formula ~ ; ~ On the normalized curve in Fig. 6, two points A and B are selected as interpolation nodes, through which the calculated curve must pass. ~ B ~B ~A A A B Fig. 6. The normalized transition characteristic of a link with transfer function 8 is equal to τ ~ e Writing the expression for points A and B, we obtain a system of two equations with two unknowns: ~ ~ A B e e Aτ b τ Resolving this system with respect to τ and T, we obtain:

31 3 ~ ~ B ln A A ln B τ ln ~ ln ~ A B A τ B τ ln ~ ln ~ A B Approximation of the transient characteristics of control objects without self-leveling by an integrating link with a delay or a real integrating link The approximating transfer function is sought in the form: W К τ e 3 or W K 4 The parameters of models 3, 4 can be easily determined by drawing the asymptote BC to the steady-state section of the acceleration curve in Fig. 6: C A α B Fig. 6. To d / d mouth gα mouth OB OA mouth 5 τ OA for model 3

32 3 TOA for model 4 Approximation of the transient characteristics of control objects by a link of nth order Since the method discussed below is intended to approximate the transient characteristics of objects without pure delay and with self-leveling, then from the acceleration curve it is necessary to first exclude the components corresponding to the links of pure delay and integrating, if there are some. To eliminate the component due to pure delay, all abscissas of the acceleration curve should be reduced by the amount of pure delay τ i.e. move the origin of coordinates to the right by τ. In this case, in the transfer function of an object with a pure delay W about We " about The AB section of the transition characteristic without delay in Fig. 7 τ " corresponds to the transition function W about. B Y A C τ A Fig.7. B α Fig.8. - When approximating the transient response of an object without self-leveling, it is represented as the difference between two characteristics in Fig. 8:

33 33 To do this, we draw the asymptote BC to the steady-state portion of the characteristic and the ray OA is parallel to BC. Subtracting from, we find. - transient characteristic of the integrating link with the transfer function W K. Coefficient K is still found according to formula 5: K gα mouth transient characteristic of an object with self-leveling. It corresponds to the transfer function W. Due to the linearity of the Laplace transform, the transfer function of the object corresponding to the characteristic is equal to: W К W W W about The coefficients of the transfer function W can be found by the method described below. Bringing the expression for W about to a common denominator, we obtain the desired transfer function of the object without self-leveling. Determining the coefficients of the transfer function of an object using the Shimoya area method The method is intended for determining the coefficients of the fractional-rational transfer function of an object of the form m bm L W about K about n 6 L n

34 34 In practice, as noted, n,3; m,. The transmission coefficient about K, as always, is determined by formula 9. To simplify the calculations, we normalize the acceleration curve of the object in the range - according to the formula. For a normalized curve ~ with a unit input action about K. Let us write the inverse expression of the transfer function 6 and expand it into an infinite series in powers of p: m n about S S S b W L 7 Reducing 7 to a common denominator and equating the coefficients at the same powers of p, we find: 8, S S b S b b S S b S b b S S b b S b L LLLLLLLLL in the special case with m S S S 9 The numerator and denominator of the desired transfer function 6 contain nm unknown coefficients, so to find them it is necessary that system 8 or in the special case 9 contain the same number equations.

35 35 So, system 8 or 9 allows you to determine the coefficients of the transfer function 6 through the still unknown expansion coefficients S. To determine the latter, consider the Laplace image of the deviation of the normalized transition response from the steady-state value: L rev ( ~ ) L() L( ~ ) [ W p ] From this we find W about ( L[ ~ ]), or taking into account the definition of the Laplace transform 3: W about [ ~ ] e d Expanding the function e into a series in powers: e!! 3 3 L L, 3!! we can represent the integral in the expression as a sum of integrals: ~ e d ~ d d ~ d! ~! ~dL! Substituting the expansions 7 and , multiplying the power series from and equating the coefficients at the same powers of p in the resulting relation, we obtain the following expressions for the coefficients S.

36 36 3!! ~, 6 ~ ~, ~, ~ d i S S d S S S S d S S S d S S d S i i i LLLLLLLLLLLLLLL In practical calculations, integrals 3 are determined by numerical methods. For example, when using the trapezoidal method, expressions for coefficients S take the form: 4.5 6 ~.5 ~.5 ~.5 ~ 3 3 ` N i i N i i N i i N i i S i i S i S S S S i i S S S S i S S S where is the interval discreteness of samples of the normalized transient response, N is the number of points of the transient response. From a geometric point of view, the coefficient S is the area bounded by the curve ~ and the line of steady values. S is the area weighted with the weighting function S, etc. Thus,

37 37 coefficients S are some weighted areas, which determines the name of the method. If during calculations the th coefficient S turns out to be negative, it is necessary in model 6 to reduce n by one or increase t i.e. reduce the difference n-m.. Industrial regulators ACP [4].. Functional diagram of an automatic regulator An automatic regulator is a set of elements that serve to regulate technological processes. The functional diagram of a closed-loop automatic control system looks like Fig. 9 ass S x Z SU FU IM RO OR IE F Automatic regulator Fig. 9. Object of regulation In Fig. 9 is indicated: Z - the adjustable variable adjuster is used to set its specified desired value; CS - comparing device, generates a mismatch signal; back FU - forming device, serves to form the regulation law in electrical regulators together with the IM; IM - actuator, activates RO;

38 38 RO - regulatory working body, serves to change the regulatory influence x; OR is actually the object of regulation; IE measuring element serves to measure the controlled variable y and convert it into a unified signal. The working body, together with the drive, if any, is usually classified as an object of regulation. The measuring element can be attributed to both the object and the controller. In cases where a measuring element is used to take a time characteristic, it is referred to as an object. Thus, an automatic regulator includes a setter of the controlled variable, a comparing device, a forming device and an actuator... Classification of regulators according to the energy consumption of an external source On this basis, regulators are divided into direct and indirect acting regulators. In direct-acting regulators, the energy of the controlled medium itself is used to rearrange the working element. For example, in a direct-acting liquid level regulator, the energy of the liquid, the level of which is regulated, is used to move the working element. Direct-acting regulators are simple and cheap, but do not provide high quality control. Their disadvantages are also the difficulty of implementing complex regulatory laws and obtaining large efforts to rearrange the working body. Indirect-acting regulators use energy from an external source to rearrange the working element, the type of which

39 39 there are electrical electronic, pneumatic, hydraulic, and combined regulators. Electric regulators have a number of advantages. Their main disadvantage in the conventional version is the impossibility of using them in fire and explosive environments. Pneumatic regulators do not have this drawback. The main advantage of hydraulic regulators is the increased power of the actuator with relatively small dimensions. Combined regulators allow you to combine the advantages of different types of regulators. For example, electro-pneumatic systems combine the advantages of electrical regulators with the ability to operate pneumatic actuators in fire and explosive environments. In recent years, programmable controllers have found widespread use for the implementation of local automation systems. The choice of regulator type is dictated by various considerations: the nature of the environment, operating conditions, special requirements..3. Classification of regulators according to the regulation law The regulation law refers to the equation of regulator dynamics. Five standard control laws are known: proportional P, integral I, proportional-integral PI, proportional-differential PD and proportional-integral-differential PID. Proportional static controllers Dynamic equation of P-regulator K 5

40 4 where is the mismatch of the controlled variable, set x is the control action more precisely, the increment of the control action relative to the constant component, therefore it is more correct to write x - x in 5 instead of x, but x is usually omitted, K is the transmission coefficient P of the regulator. As we see from 5, the regulating effect of the P regulator is proportional to the mismatch, i.e. The P controller is an inertia-free link with the transfer function W K. Since the P controller does not introduce a negative phase shift in the phase response of the P controller into the system, the ASR with the P controller has ϕ good dynamic properties. The disadvantage of systems with a P regulator is the presence of a static error. For a single controller, the magnitude of this error is determined from the controller equation: K When the P controller operates in the system Fig. F K K about Fig.. the magnitude of the error from the disturbance F is

41 4 FК ЗСF F К about Kob К р, where perturbation. K ZCF - transmission coefficient of a closed-loop system As we can see, the static error in a system with a P controller is inversely proportional to its transmission coefficient, the limiting value of which is determined by the required stability margin of a closed-loop ASR. Proportional controllers are used in the automation of low-inertia control objects, when the value of K can be selected by error. large enough to reduce the static Integral astatic regulators Regulation law: K d, 6 i.e. the regulatory effect in this case is proportional to the integral of the mismatch. The transfer coefficient of the I-regulator K d / d has the meaning of the rate of change of the regulatory action per unit of mismatch. Transfer function: K W Frequency transfer function:

42 4 K K W j j e The advantage of the AND controller is zero static error. From 6 it follows that this error is equal and in statics it becomes zero. d / d K At the same time, since the phase response of the AND controller is ϕ π, the system with the AND controller has very poor dynamic properties, because this regulator introduces a negative phase shift π into the system. Integral regulators can only be used when automating practically inertia-free objects. ASR with both a regulator and an object without self-leveling is structurally unstable, π j i.e. unstable at any controller settings. Proportional integral regulators The control law of a PI regulator can be written in two forms: K K d K d 7 T The regulatory effect of a PI regulator represents the sum of the P and I components with proportionality coefficients K and K. From a comparison of the two forms of writing the control law, we obtain: K , K T I I

43 43 where T and isodrome time. K >> Transfer function and frequency transfer function: W W K j K K K, K e I K jrcg K From the last expression it is clear that in the region of low frequencies at K PI the regulator behaves like an AND regulator. At high K frequencies K >>, i.e. The PI controller behaves like a P controller. This allows the PI regulator to combine the advantages of an I regulator in statics and a P regulator in dynamics. The physical meaning of the isodrome time can be explained by the transient characteristic of the PI controller in Fig. As can be seen from this figure, TI is the doubling time of the P component of the regulating influence of the PI regulator, or, what is the same, the time by which the regulating influence of the PI regulator is ahead of the regulating influence of the I regulator. The value TI characterizes the speed of integration. The higher the TI, the lower the integration speed. With T and PI, the regulator turns into a P regulator. K x PI I K P I Fig..

44 44 So, an ASR with a PI regulator has zero static error due to the presence of the AND component in the control law. This is true for all regulators with an AND component. As can be seen from the phase response of the PI regulator, Fig., in the region of working 3 ϕ slave π Fig.. slave frequencies, the PI regulator introduces a negative phase shift of approximately -3 into the system. This is significantly less than the I regulator, but more than the P regulator. Therefore, the dynamic properties of an ASR with a PI regulator are much better than with an I-regulator, but worse than with a P regulator. Proportional - differential regulators The regulation law of an ideal PD regulator: d d K K K P, 8 d d where K, K are the proportionality coefficients of the P- and D- components of the regulation law. T P anticipation time. Transfer and frequency transfer functions: W W K K j K K K e P, K jrcg K

45 45 From the last expression it is clear that at low frequencies the PD regulator behaves like a P regulator, and at high frequencies it behaves like a differentiator. Since an ideal differentiating link is physically impossible, real PD controllers use a real inertial differentiating link. The transfer function of such a controller has the form W K K The smaller the time constant T, the closer the characteristics of the ideal and real controllers. In statics, the transfer function of the PD controller coincides with the transfer function of the P-regulator; therefore, ASR with a PD controller also has a static error. As can be seen from the phase response curve in Fig. 3, ϕ π ideal -3 real slave Fig. 3. in the region of operating frequencies, the PD regulator introduces a positive phase shift into the system, increasing its stability margin. Therefore, an ASR with a PD regulator has the best dynamic properties. For the same reason, the value of K can be chosen greater than in the case of P

46 46 regulator. Therefore, the static error in an ASR with a PD controller is less than in a system with a P controller. However, PD regulators are practically not used, because in the presence of high-frequency interference superimposed on a low-frequency useful signal, the differentiation operation sharply worsens the signal-to-noise ratio, as a result of which the amplitude of the noise derivative can significantly exceed the amplitude of the useful signal derivative. Regarding the physical meaning of the advance time, we can say that T P is the time by which the regulating effect of the PD regulator advances the regulating effect of the P regulator with a linear input effect Fig. 4 x PD PD P Fig. 4. Proportional - integral differential controllers Dynamic equation: d d К К d К К d П d 9 d И Transfer functions of ideal and real PID controllers:

47 47 W W K K K K K K K K I P, Frequency transfer function of an ideal PID controller: W j K K K e K K jrcg K Systems with PID controllers combine zero static error with good dynamics, since, as can be seen from the phase response of the PID controller in Fig. .5 in the area of ​​operating frequencies the PID controller is the same as ϕ π ideal slave real π Fig. 5. and P regulator does not introduce a negative phase shift into the system. To increase the noise immunity of the PID controller in practice, the ratio of advance time/isodrome time is limited from above by the inequality / PI<,5, 3 поэтому помехоустойчивость ПИД регулятора выше, чем ПД регулятора. При выборе закона регулирования учитывают следующие соображения.

48 48 If a static error is unacceptable, the controller must contain an AND component. In order of deterioration of dynamic properties, the control laws are arranged in the following order: PD, PID, P, PI, I. Regulators with a D component have poor noise immunity. For this reason, PD regulators are practically not used, and PI regulators are used with limitation 3. PI and PID control laws are most widely used in practice. 3. Calculation of regulator settings in linear continuous systems [4] 3.. Quality of regulation We will determine the quality of regulation by a set of indicators characterizing the shape of the transition process curve in a closed ASR Fig. 6. Main quality indicators. The maximum dynamic deviation dyn is the largest deviation of the controlled variable from its set value in the transient process. Indicator dyn m set. In a stable automatic control system, the first deviation is the maximum. din characterizes the dynamic accuracy of regulation. Residual deviation residual unevenness ct - absolute static regulation error, defined as the difference between the steady-state value of the controlled variable and its specified value:

49 49 st mouth ass Indicator in static mode. m st characterizes the accuracy of regulation in mouth back din 3 δ st Fig. The degree of attenuation ψ is the ratio of the difference between two adjacent oscillation amplitudes directed on one side of the line of the steady value to the larger of them 3 3 ψ ;< ψ < 3 Показатель ψ характеризует колебательность переходных процессов и запас устойчивости системы. Значение ψ соответствует незатухающим колебаниям на границе устойчивости системы. При ψ имеем апериодический переходной процесс. 4. Время регулирования промежуток времени от момента нанесения возмущающего воздействия до момента, начиная с которого отклонение регулируемой переменной от установившегося значения становится и остается меньше наперёд заданного значения δ. Показатель характеризует быстродействие системы.

50 5 The considered quality indicators belong to the group of direct indicators, i.e. indicators that allow you to assess the quality directly from the transition process curve, to obtain which it is necessary to solve the differential equation of the system. In addition to direct ones, there are indirect criteria that allow one to judge the quality of regulation without having the transition process curve at their disposal. Such criteria, in particular, include integral quality criteria, representing integrals over time of the deviation of the controlled variable from the steady-state value, or of some function of this deviation and its derivatives. The simplest is the linear integral criterion determined by the relation: I line d mouth From a geometric point of view, the criterion I line is the area between the curve and the mouth line. The value of I lin depends on all quality indicators, except Art. At the same time, with a decrease in dyne, etc. As the quality of regulation improves, the value of Ilin decreases, and with an increase in the oscillation of the transient process, Ilin also decreases, although the quality of regulation deteriorates. So, a decrease in I lin indicates an improvement in the quality of regulation only for well-damped transient processes. Therefore, the I lin criterion is applicable for aperiodic or weakly oscillatory processes. For such processes, the best regulator settings can be considered those at which the value of Ilin reaches a minimum. The Ilin criterion can be calculated through the coefficients of the differential equation of a closed-loop ASR.

51 5 It can be shown that for a control object with self-leveling and a PI regulator I line, 3 K i.e. the minimum I lin is achieved at the maximum integral component of the regulatory action, or, what is the same, the best quality of the transient process is achieved at maximum K. For oscillatory transient processes, other integral criteria are used, for example, I mode mouth d, but this criterion cannot be calculated through the coefficients of the differential equations The quadratic integral criterion I quarter is free from this drawback: I quarter mouth d 3.. Typical optimal processes Requirements for quality indicators are contradictory. For example, reducing the dynamic error is achieved by increasing the oscillation and duration of transient processes. On the contrary, processes with short control times can be achieved by increasing the dynamic error. Therefore, a compromise decision has to be made regarding the desired values ​​of quality indicators in a closed-loop automated control system. Transient processes with certain quality indicators are recommended as standard ones when calculating ACP. In the extended frequency method discussed below

52 5 characteristics the main indicator of quality is the degree of attenuation ψ, i.e. oscillation of the transition process, since this indicator characterizes the stability margin of the ASR. Processes for which ψ,75.9 are recommended as typical, i.e. the third vibration amplitude is 4 times less than the first. In cases where the task is to select controller settings that minimize any quality indicator, the corresponding transient process, as well as the values ​​of the controller settings, are called optimal in the sense of the specified criterion. For example, in the method of extended frequency characteristics, the task is to select the controller settings in such a way that, in addition to the specified oscillation of the transient process, the minimum value of the I lin criterion is ensured. Such a process is optimal in the sense of the I lin criterion. Simplified formulas for calculating regulator settings In Table. simplified formulas are given for determining the settings of the regulators that provide the specified oscillation of the transient process. The formulas are obtained from the results of ASR modeling. Static objects are represented by a model of an inertial link with a pure delay of 8, astatic objects by a model of an integrating link with a delay of 3

Lecture 3 Mathematical description of control systems In control theory, when analyzing and synthesizing control systems, we deal with their mathematical model. The mathematical model of the automatic control system is the equation

Test 1 in the discipline “Management of technical systems” Option 1 1. What is the functional purpose of the sensor in the control system? 1) regulate the parameters of the technological process; 2) suppress noise

Equations of dynamics and statics. Linearization At a certain stage of development and research of an automatic control system, its mathematical description is obtained; a description of the processes occurring in the system

METHODOLOGICAL INSTRUCTIONS for homework for the course of technical control Research of a nonlinear automatic control system DETERMINATION OF INITIAL DATA Initial data for completing homework are given

Fundamentals of Control Theory Doctor of Technical Sciences Mokrova Natalia Vladislavovna Dynamic characteristics of regulated objects 1. Temporal characteristics. Acceleration curve. Pulse transient function. 2. Solution of differential

FSBEI HPE "Omsk State Technical University" SECTION II CONTINUOUS LINEAR AUTOMATIC CONTROL SYSTEMS Lecture 4. DYNAMIC LINKS. GENERAL CONCEPTS, TIME CHARACTERISTICS AND FREQUENCY

Practical lesson TRANSFER FUNCTION FREQUENCY CHARACTERISTICS Goals and objectives of the work As a result of mastering the topic, the student should be able to obtain an operator equation from a given differential equation;

Lecture 5 Automatic regulators in control systems and their configuration Automatic regulators with standard control algorithms: relay, proportional (P), proportional-integral (PI),

Calculation of the dynamic characteristics of linear automatic control systems Determine the weight function g(t) and the transition function h(t) of a linear automatic control system, consisting of a series connection of an aperiodic and an ideal integrating

Lecture 3. Mathematical description of control objects 1. Control objects In the chemical industry, typical control objects include various processes in the apparatus of technological installations. For

Lecture 8 33 ONE-DIMENSIONAL STATIONARY SYSTEMS APPLICATION OF THE FOURIER TRANSFORM 33 Description of signals and systems Description of signals To describe deterministic signals, the Fourier transform is used: it

Federal State Budgetary Educational Institution of Higher Professional Education KAZAN NATIONAL RESEARCH TECHNICAL UNIVERSITY named after. A.N.TUPOLEVA-KAI Department of Television

Lecture 4 Typical dynamic links It is convenient to represent automatic control systems as a connection of elements, each of which is described by an algebraic or differential equation

LABORATORY WORK 5 TYPICAL UNITS OF AUTOMATIC SYSTEMS The purpose of the work is to study the dynamic properties of typical units of automatic control systems GENERAL INFORMATION In the theory of automatic control

Lecture 11.12 Section 2: MATHEMATICAL MODELS OF LINEAR CONTROL SYSTEMS Topic 2.4: TYPICAL DYNAMIC UNITS OF SYSTEMS 1. Typical units of systems: characteristics and equations; physical models. Lecture outline:

UDC: 62-529 AUTOMATIC CONTROL SYSTEMS WITH SEQUENTIAL CORRECTION Vitaly Anatolyevich Chigarev Senior Lecturer, Belarusian National Technical University, [email protected]

Topic 8 LINEAR DISCRETE SYSTEMS Concept of discrete system Methods for describing linear discrete systems: difference equation, transfer function, impulse response, frequency transfer function

Continuous-deterministic models Continuous-deterministic models are used for the analysis and design of dynamic systems with continuous time, the process of functioning of which is described

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Autonomous Educational Institution of Higher Education "NATIONAL RESEARCH TOMSK POLYTECHNIC UNIVERSITY"

Topic 3 HARMONIC ANALYSIS OF NON-PERIODIC SIGNALS Direct and inverse Fourier transforms Spectral characteristics of a signal Amplitude-frequency and phase-frequency spectra Spectral characteristics

Autumn semester of the academic year Topic 3 HARMONIC ANALYSIS OF NON-PERIODIC SIGNALS Direct and inverse Fourier transforms Spectral characteristics of the signal Amplitude-frequency and phase-frequency spectra

4. TRANSITION CHARACTERISTICS OF THE MEMBRANE 4.1 Temporary characteristics of a dynamic system To assess the dynamic properties of the system and individual links, it is customary to study their response to typical input influences,

64 Lecture 6 OPERATOR METHOD OF ANALYSIS OF ELECTRIC CIRCUITS Plan Laplace transform Properties of the Laplace transform 3 Operator method of analysis of electrical circuits 4 Determination of the original from the known

Seminar 4. ANALYSIS OF SELF-OSCILLATIONS BY THE HARMONIC LINEARIZATION METHOD Statement of the problem A closed system with one nonlinear element is considered. g F (z W (s x Fig. The free movement of the system is studied,

Federal Agency for Education State educational institution of higher professional education Vladimir State University Department of plastics processing technology UDC

Completed: Accepted by: Umarov D. 1-14 IKSUTP Abdurakhmanova M.I. Analysis of the stability of ACS The practical suitability of control systems is determined by their stability and acceptable quality of control. Under

54 Lecture 5 FOURIER TRANSFORM AND SPECTRAL METHOD FOR ANALYSIS OF ELECTRICAL CIRCUITS Plan Spectra of aperiodic functions and the Fourier transform Some properties of the Fourier transform 3 Spectral method

1. Automatic regulation of the water level in the steam generator Regulation of the power supply in each of the steam generators (SG) comes down to maintaining a material balance between steam removal, purge and supply

Mathematical schemes: D-schemes Continuously deterministic models are used for the analysis and design of dynamic systems with continuous time, the process of functioning of which is described by deterministic

4.1 Test questions for self-control 1 SECTION “Linear continuous models and characteristics of control systems” 1 What does control theory study? 2 Define the concepts of management and control object.

Lecture 5. 8.3. ANALYSIS OF SELF-OSCILLATIONS BY THE METHOD OF HARMONIC LINEARIZATION 8.3.. Statement of the problem A closed system with one nonlinear element is considered. F W s x Fig. Free movement is being studied

Institute Direction of training AVTI 70404 Management in technical systems Bank of assignments for a special part of the entrance test for a master's degree Examination paper 6 (5 points) Topic

Topic 8 DISCRETE ACS Lecture 7 General concepts and definitions of the theory of discrete ACS. Basic information about the mathematical apparatus of the theory of linear discrete stationary systems. Mathematical description of processes

Lecture 4 Frequency characteristics of ACS systems Frequency characteristics of ACS characterize the response of systems to a sinusoidal input influence in a steady state. Frequency characteristics include:

THEORY OF STABILITY OF LINEAR SYSTEMS 1. Basic terms and definitions Any ACS is always subject to external disturbances that can disrupt its normal operation. A properly designed self-propelled gun should

Lecture 1 General information about control systems The subject “Theory of automatic control” introduces you to the basic principles of constructing automatic control systems, methods of formalized description

Guidelines for laboratory work in the course “Theory of Automatic Control” Module “Linear Automatic Systems” Laboratory work Determination of parameters of typical dynamic links

Robotics RAR1300 Sergei Pavlov TTÜ Virumaa Kolledž Drive control Control of the movement of a working machine or mechanism means controlling the position, speed and acceleration of a system that

TAU Practical exercises Assignments for test work and methodological instructions for its implementation Practical lesson AFFC, LAX, transition and weight characteristics of typical dynamic links Most

Lecture 6 CIRCUITS OF PERIODIC NON-SINUSIDAL CURRENT Plan Trigonometric form of the Fourier series Fourier series in complex form Complex frequency spectrum 3 Powers in non-sinusoidal current circuits Coefficients,

SEMINAR Basic concepts. Drawing up (derivation) of a differential equation. The concept of solving a differential equation. Solution by the method of separable variables. Solving a Linear Differential Equation

Fundamentals of circuit design FUNDAMENTALS OF CIRCUIT DESIGN...1 1. BASIC PROVISIONS...1 2. Amplification OF WEAK SIGNALS...6 3. Amplification OF STRONG SIGNALS...14 4. FUNDAMENTALS OF AMPLIFIER CIRCUIT DESIGN...18 1. Fundamentals

Fundamentals of Control Theory Doctor of Technical Sciences Mokrova Natalia Vladislavovna Lecture 7 Nonlinear automatic control systems Features of nonlinear systems. Typical nonlinearities of automatic control systems.

Lecture 4 Frequency functions and characteristics 4 The concept of frequency functions and characteristics An important role in the study of linear stationary systems is played by frequency characteristics. They represent

70 Lecture 7 OPERATOR FUNCTIONS OF CIRCUITS Plan Operator input and transfer functions Poles and zeros of circuit functions 3 Conclusions Operator input and transfer functions The operator function of a circuit is called

I Study of the dynamics of typical automation links 1 Ideal amplifier (zero-order aperiodic link - AP-0) and a real amplifier (first-order aperiodic link - AP-1) Purpose of the work: to study

Setting up and adjusting automatic regulators. 1.Special cycle 1.1. Introduction Main stages and dates in the development of automatic control. Until 1600 Control system consisting of a float

Laboratory work 1 1 DYNAMIC CHARACTERISTICS OF TYPICAL UNITS 1. Purpose of the work To study the dynamic characteristics of typical units of automatic control systems (ACS), and also to get acquainted

Ministry of Education of the Republic of Belarus Educational institution Belarusian State University of Informatics and Radioelectronics Department of Radio Engineering Systems Laboratory work report “RESEARCH”

1. GENERAL INFORMATION ABOUT ANALOG ELECTRONIC DEVICES (AED). PARAMETERS AND CHARACTERISTICS OF AED 1. 1. General information about analog electronic devices (AED), principles of their construction Analog signals

Laboratory work 1 1 TYPICAL UNITS OF ACS 1. Purpose of the work To study the dynamic characteristics of typical units of automatic control systems (ACS), as well as to get acquainted with the basic rules of structural

Topic 5 LINEAR STATIONARY SYSTEMS Properties of linear stationary systems: linearity, stationarity, physical realizability Differential equation Transfer function Frequency transfer function

Lecture 6 Transformation of mathematical models of systems. Transfer functions. Models in the form of signal graphs To study the properties of complex physical systems and learn to control them, you must have

UDC 681.52 ALGORITHMS FOR SOLVING THE IDENTIFICATION PROBLEM N.V. Plotnikova, N.S. Kalistratova, O.N. Malyavkin Recently, due to the increasingly high demands placed on management processes in various

Topic 2. Basic concepts and definitions in the theory and practice of automatic regulation of life support parameters (2 hours) In order to ensure the normal operation of the controlled object (OR)

54 Lecture 5 FOURIER TRANSFORM AND SPECTRAL METHOD FOR ANALYSIS OF ELECTRIC CIRCUITS Plan Spectra of aperiodic functions and the Fourier transform 2 Some properties of the Fourier transform 3 Spectral method

Zaitsev G. F. Theory of automatic control and regulation Second edition, revised and expanded Approved by the Ministry of Higher and Secondary Special Education of the USSR as a teaching aid

1.1. Methods for analyzing nonlinear-inertial properties of analog devices In the literature devoted to the analysis of nonlinear-inertial properties of analog devices, several